Optimal model reduction by empirical spectral methods via sampling of chaotic orbits

Proper orthogonal decomposition (POD) coupled with the Galerkin method is applied to the onedimensional model of a tubular reactor with external heat recycle, which exhibits periodic and chaotic solutions. The effect of the cooling medium temperature on the system dynamics is considered. The issue of the optimal construction of the POD basis is addressed by sampling of the chaotic orbits, with the aim of constructing a global basis for a reduced-order model (ROM). To demonstrate that such orbits are the most appropriate because they incorporate the maximum amount of information about the system behavior, the entropy of the orbit is calculated. Sampling of the chaotic solutions allows for the determination of the POD basis to be employed in the POD/Galerkin method. The accuracy of the ROMs is compared by means of the Hausdorff distance, computed between the asymptotic regime of the reference solution and each of the ROM-reconstructed asymptotic attractors. A norm computed on the sampled time series is employed to compare transient solutions. The POD-based ROMs work well even for values of the parameter for which the model behavior is far from chaotic, i.e. periodic orbits or fixed points. Moreover, the POD-based ROMs successfully compare with a classic orthogonal collocation method.